 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dqrt02()

 subroutine dqrt02 ( integer M, integer N, integer K, double precision, dimension( lda, * ) A, double precision, dimension( lda, * ) AF, double precision, dimension( lda, * ) Q, double precision, dimension( lda, * ) R, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( lwork ) WORK, integer LWORK, double precision, dimension( * ) RWORK, double precision, dimension( * ) RESULT )

DQRT02

Purpose:
``` DQRT02 tests DORGQR, which generates an m-by-n matrix Q with
orthonornmal columns that is defined as the product of k elementary
reflectors.

Given the QR factorization of an m-by-n matrix A, DQRT02 generates
the orthogonal matrix Q defined by the factorization of the first k
columns of A; it compares R(1:n,1:k) with Q(1:m,1:n)'*A(1:m,1:k),
and checks that the columns of Q are orthonormal.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix Q to be generated. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix Q to be generated. M >= N >= 0.``` [in] K ``` K is INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0.``` [in] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) The m-by-n matrix A which was factorized by DQRT01.``` [in] AF ``` AF is DOUBLE PRECISION array, dimension (LDA,N) Details of the QR factorization of A, as returned by DGEQRF. See DGEQRF for further details.``` [out] Q ` Q is DOUBLE PRECISION array, dimension (LDA,N)` [out] R ` R is DOUBLE PRECISION array, dimension (LDA,N)` [in] LDA ``` LDA is INTEGER The leading dimension of the arrays A, AF, Q and R. LDA >= M.``` [in] TAU ``` TAU is DOUBLE PRECISION array, dimension (N) The scalar factors of the elementary reflectors corresponding to the QR factorization in AF.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK.``` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (M)` [out] RESULT ``` RESULT is DOUBLE PRECISION array, dimension (2) The test ratios: RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS ) RESULT(2) = norm( I - Q'*Q ) / ( M * EPS )```

Definition at line 133 of file dqrt02.f.

135*
136* -- LAPACK test routine --
137* -- LAPACK is a software package provided by Univ. of Tennessee, --
138* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
139*
140* .. Scalar Arguments ..
141 INTEGER K, LDA, LWORK, M, N
142* ..
143* .. Array Arguments ..
144 DOUBLE PRECISION A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
145 \$ R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
146 \$ WORK( LWORK )
147* ..
148*
149* =====================================================================
150*
151* .. Parameters ..
152 DOUBLE PRECISION ZERO, ONE
153 parameter( zero = 0.0d+0, one = 1.0d+0 )
154 DOUBLE PRECISION ROGUE
155 parameter( rogue = -1.0d+10 )
156* ..
157* .. Local Scalars ..
158 INTEGER INFO
159 DOUBLE PRECISION ANORM, EPS, RESID
160* ..
161* .. External Functions ..
162 DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
163 EXTERNAL dlamch, dlange, dlansy
164* ..
165* .. External Subroutines ..
166 EXTERNAL dgemm, dlacpy, dlaset, dorgqr, dsyrk
167* ..
168* .. Intrinsic Functions ..
169 INTRINSIC dble, max
170* ..
171* .. Scalars in Common ..
172 CHARACTER*32 SRNAMT
173* ..
174* .. Common blocks ..
175 COMMON / srnamc / srnamt
176* ..
177* .. Executable Statements ..
178*
179 eps = dlamch( 'Epsilon' )
180*
181* Copy the first k columns of the factorization to the array Q
182*
183 CALL dlaset( 'Full', m, n, rogue, rogue, q, lda )
184 CALL dlacpy( 'Lower', m-1, k, af( 2, 1 ), lda, q( 2, 1 ), lda )
185*
186* Generate the first n columns of the matrix Q
187*
188 srnamt = 'DORGQR'
189 CALL dorgqr( m, n, k, q, lda, tau, work, lwork, info )
190*
191* Copy R(1:n,1:k)
192*
193 CALL dlaset( 'Full', n, k, zero, zero, r, lda )
194 CALL dlacpy( 'Upper', n, k, af, lda, r, lda )
195*
196* Compute R(1:n,1:k) - Q(1:m,1:n)' * A(1:m,1:k)
197*
198 CALL dgemm( 'Transpose', 'No transpose', n, k, m, -one, q, lda, a,
199 \$ lda, one, r, lda )
200*
201* Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
202*
203 anorm = dlange( '1', m, k, a, lda, rwork )
204 resid = dlange( '1', n, k, r, lda, rwork )
205 IF( anorm.GT.zero ) THEN
206 result( 1 ) = ( ( resid / dble( max( 1, m ) ) ) / anorm ) / eps
207 ELSE
208 result( 1 ) = zero
209 END IF
210*
211* Compute I - Q'*Q
212*
213 CALL dlaset( 'Full', n, n, zero, one, r, lda )
214 CALL dsyrk( 'Upper', 'Transpose', n, m, -one, q, lda, one, r,
215 \$ lda )
216*
217* Compute norm( I - Q'*Q ) / ( M * EPS ) .
218*
219 resid = dlansy( '1', 'Upper', n, r, lda, rwork )
220*
221 result( 2 ) = ( resid / dble( max( 1, m ) ) ) / eps
222*
223 RETURN
224*
225* End of DQRT02
226*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine dsyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
DSYRK
Definition: dsyrk.f:169
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
Definition: dorgqr.f:128
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
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